a)
\frac{2+\sqrt5}{\sqrt5-1}=\frac{(2+\sqrt5)(\sqrt5+1)}{(\sqrt5-1)(\sqrt5+1)}=\frac{2\sqrt5+2+5+\sqrt5}{5-1}=\frac{7+3\sqrt5}{4}
b)
\frac{3-\sqrt2}{\sqrt2+2}=\frac{(3-\sqrt2)(\sqrt2-2)}{(\sqrt2+2)(\sqrt2-2)}=\frac{3\sqrt2-6-2+2\sqrt2}{2-4}=\frac{-8+5\sqrt2}{-2}=\frac{8-5\sqrt2}{2}
c)
\frac{\sqrt3+4}{2\sqrt3+2}=\frac{(\sqrt3+4)(2\sqrt3-2)}{(2\sqrt3+2)(2\sqrt3-2)}=\frac{2*3-2\sqrt3+8\sqrt3-8}{4*3-4}=\frac{6\sqrt3-2}{8}=\frac{2(3\sqrt3-1)}{8}=\frac{3\sqrt3-1}{4}
e)
\frac{\sqrt2+3}{3-\sqrt2}=\frac{(\sqrt2+3)(3+\sqrt2)}{(3-\sqrt2)(3+\sqrt2)}=\frac{(\sqrt2+3)^2}{9-2}=\frac{2+6\sqrt2+9}{7}=\frac{11+6\sqrt2}{7}
f)
\frac{4-2\sqrt3}{4+2\sqrt3}=\frac{(4-2\sqrt3)(4-2\sqrt3)}{(4+2\sqrt3)(4-2\sqrt3)}=\frac{(4-2\sqrt3)^2}{16-4*3}=\frac{16-16\sqrt3+4*3}{4}=\frac{28-16\sqrt3}{4}=\frac{4(7-4\sqrt3)}{4}=7-4\sqrt3